If the person knowingly gives false information to the police, they can be charged with an additional offense. The charge may be intent to defraud, making a false statement to a police officer, or something similar. This charge is in addition to any other charges regarding the accident, including failure to provide proof of insurance.
With most of the United States participating in a national database of insurance eligibility, claiming to be insured when you are not is rarely effective, and may not be charged as a criminal act in itself if the officer perceives no intentional deceit, such as having an outdated insurance card which you believe to still be valid. The best course of action is to be honest with the officer and tell him that your insurance has expired or explain your case reasonably and truthfully.
Deliberately falsifying the course of events, such as claiming someone else was driving when they weren't, is a misdemeanor crime in itself and result in arrest. Don't embellish the story or try to paint a picture that conflicts with the facts. Law enforcement officers are trained to objectively look at an incident, and many times the statements from those involved are simply meant to fill in the fine details, not to establish who caused the accident or why it happened.